Substrate curvature resulting from the capillary forces of a liquid drop

Original entry: Sorell Massenburg, APPHY 226, Spring 2009

F. Spaepen, J. Mech. Phys. Solids, 44, 675 – 681, 1996.

Abstract

The difference between interfacial tension and interface stress is illustrated by considering a hemispherical liquid drop on a solid substrate. The equilibrium shape is determined by minimizing the total interfacial free energy, which leads to the Young equation for balance of the interfacial tensions. The curvature of the substrate is determined by the interfacial stresses. Two contributions are calculated: one arising from the hydrostatic pressure in the drop, the other from the imbalance of the interfacial stresses.

Keywords

Diagram of a droplet on a substrate. R is the radius of curvature of the droplet. $\theta$

A droplet of liquid on the surface of a solid is presumed to make a circular interface while the liquid-vapor (lv) interface is presumed to semi-circular (the reviewer presumes that this holds for all droplets that are small enough). The radius of curvature of the lv interface is denoted by R. The angle, $\theta$ cut out by making a straight line of distance, R, from the both ends of the droplet to the center is considered to be 2$\theta$

The change in the free energy of the surface is simply the difference between the sum solid-liquid (sl) with lv surface energies and the solid-vapor (sv) surface energy (final-initial surface energy). $\Delta E_{free}=A_{sl}\gamma _{sl}+A_{lv}\gamma _{lv}-A_{sl}\gamma _{sv}=2R \left \{sin(\theta)\gamma _{sl}+\theta\gamma) _{lv} \right \}-2R\gamma _{sv}$.

After minimizing vialagrange mulipliers (with the constraint of constant volume: $R^2(\theta-sin\theta cos\theta)$ (sic) and some other unpleasant arithmetic, Prof. Spaepen gets to the well known result: $cos(\theta)=-\frac{\gamma _{sl}-\gamma _{sv}}{\gamma _{lv}}$. The value of the lagrange multiplier is: $\lambda = \frac{\gamma _{lv}}{R}$ or the pressure difference across the curved interface.

Diagram of from interfacial tensions arising from a droplet on a substrate.
Diagram of lv interface indicating pressure difference and relevant quantities.

The stress in the droplet associated with pressure of the droplet is hydrostatic, is equal to: $\Delta p = \frac{\gamma _{lv}}{R}$

Free body diagram of the vertical forces on the substrate in contact with a droplet.

The hydrostatic pressure gives rise to the downward force, $\Delta p2Rsin(\theta)$ that balances the vertical components of the capillary force from the interfacial tension, $2\gamma _{lv}$. The spacing of the two capillary forces (in a 2D world) is: $L=2Rsin(\theta)$. It is simple to balance the forces and then that is used to obtain the strain of the surface: $\epsilon _0=\frac{6Fx}{Et^2}\left (\frac{x}{L}-1\right )$, where t is the substrate thickness. Integrating the strain over the L gives the elongation of the "top fiber", $\Delta L = -\frac{FL^2}{Et^2}$. Then the average curvature of the substrate is just which the ratio of twice the total tip elongation to the effected substrate thickness $\kappa_1=\frac{2\Delta}{tL}=-\frac{4\gamma _{lv}Rsin^2(\theta)}{t^3E}$.

Free body diagram with horizontal capillary forces when the horizontal force balance is off center.

When center of the force balance is off center (but the reviewer assumes that the droplet is still symmetrical). Then in order for the net curvature to be zero outside of the droplet area, then the interfacial stress at the vapor surface must be identical to the must be identical at the top and bottom. This implies a constant curvature which is given by the Stoney equation: $\kappa_2=\frac{6(f_{sv}-f_{sl}-\gamma_{sl} cos \theta )}{t^2E}$ The result is engendered by the consideration of interfacial stresses. If interfacial tensions were used, then the terms of the parentheses would be zero via the Young equation.